Question

Solve the problem. Let H be the set of all polynomials having degree at most 4 and rational coefficients. Determine whether His a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy A: Contains zero vector B: Closed under vector addition C Closed under multiplication by scala His not a vector space, not dosed under multiplication to scalars His a vector space His not a vector space:not closed under vector addition O H is not a vector space does not contain zece vector Moving to another question will save this response

Answer 1

Solution:

Given that

$H=\left \{ a+bx+cx^{2}+dx^{3}+ex^{4}:a,b,c,d,e \,\varepsilon Q \right \}$

(a) H contains Zero vectors.

(b)

H is closed under vector addition

$\therefore$ Sum of two rational no is again a a rational number.

Let,

P1(x) = (1 + b 2 +.2.2 + d2.23 +01.24

$P_{2}(x)=a_{2}+b_{2}x+c_{2}x^{2}+d_{2}x^{3}+e_{2}x^{4}$

$P_{1}(x)+P_{2}(x)=(a_{1}+a_{2})+(b_{1}+b_{2})x+(c_{1}+c_{2})x^{2}+(d_{1}+d_{2})x^{3}+(e_{1}+e_{2})x^{4}$$\Rightarrow P_{1}(x)+P_{2}(x)\epsilon H$

(c)

H is not closed under multiplication by scalars.

Let,

$P(x)=1+2x+\frac{3}{2}x^{2}+\frac{1}{2}x^{3}+5x^{4}$

Let  $\alpha =\sqrt{2}$

$\alpha P(x)=\sqrt{2}+2\sqrt{2}x+\frac{3}{\sqrt{2}}x^{2}+\frac{1}{\sqrt{2}}x^{3}+5\sqrt{2}x^{4}$

all the coefficients does not belong to Q.

$\Rightarrow$ It is not closed under scalar multiplication.